Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares

Let \(S\) be a connected closed oriented surface of genus \(g\). Given a triangulation (resp. quadrangulation) of \(S\), define the index of each of its vertices to be the number of edges originating from this vertex minus \(6\) (resp. minus \(4\)). Call the set of integers recording the non-zero indices the profile of the triangulation (resp. quadrangulation). If \(\kappa\) is a profile for triangulations (resp. quadrangulations) of \(S\), for any \(m\in \mathbb{Z}_{>0}\), denote by \(\mathscr{T}(\kappa,m)\) (resp. \(\mathscr{Q}(\kappa,m)\)) the set of (equivalence classes of) triangulations (resp. quadrangulations) with profile \(\kappa\) which contain at most \(m\) triangles (resp. squares). In this paper, we will show that if \(\kappa\) is a profile for triangulations (resp. for quadrangulations) of \(S\) such that none of the indices in \(\kappa\) is divisible by \(6\) (resp. by \(4\)), then \(\mathscr{T}(\kappa,m)\sim c_3(\kappa)m^{2g+|\kappa|-2}\) (resp. \(\mathscr{Q}(\kappa,m) \sim c_4(\kappa)m^{2g+|\kappa|-2}\)), where \(c_3(\kappa) \in \mathbb{Q}\cdot(\sqrt{3}\pi)^{2g+|\kappa|-2}\) and \(c_4(\kappa)\in \mathbb{Q}\cdot\pi^{2g+|\kappa|-2}\). The key ingredient of the proof is a result of J. Kollár on the link between the curvature of the Hogde metric on vector subbundles of a variation of Hodge structure over algebraic varieties, and Chern classes of their extensions. By the same method, we also obtain the rationality (up to some power of \(\pi\)) of the Masur-Veech volume of arithmetic affine submanifolds of translation surfaces that are transverse to the kernel foliation.